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Simplify. Rationalize the denominator.\newline3−2−5\frac{3}{-2 - \sqrt{5}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline3−2−5\frac{3}{-2 - \sqrt{5}}
  1. Identify Conjugate of Denominator: Identify the conjugate of the denominator.\newlineThe conjugate of a−ba - \sqrt{b} is a+ba + \sqrt{b}. Therefore, the conjugate of −2−5-2 - \sqrt{5} is −2+5-2 + \sqrt{5}.
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator.\newline3×(−2+5)(−2−5)×(−2+5)\frac{3 \times (-2 + \sqrt{5})}{(-2 - \sqrt{5}) \times (-2 + \sqrt{5})}
  3. Apply Difference of Squares: Apply the difference of squares formula to the denominator.\newlineThe difference of squares formula is (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2. Applying this to the denominator we get:\newline(−2)2−(5)2=4−5=−1(-2)^2 - (\sqrt{5})^2 = 4 - 5 = -1.
  4. Distribute Numerator: Distribute the numerator.\newlineMultiply 33 by each term in the conjugate.\newline3×(−2)+3×5=−6+35.3 \times (-2) + 3 \times \sqrt{5} = -6 + 3\sqrt{5}.
  5. Combine and Simplify: Combine the results and simplify.\newlineSince the denominator simplifies to −1-1, we divide the entire numerator by −1-1 to get the final simplified form.\newline(−6+35)/−1=6−35(-6 + 3\sqrt{5}) / -1 = 6 - 3\sqrt{5}.

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